Diffraction grating equation for order positions Steps:

• Slit spacing d = 1 / (400 lines/mm × 1000 mm/m) = 2.5 × 10^{-6} m.
• λ/d = 587 × 10^{-9} / 2.5 × 10^{-6} = 0.2348.
• θ₂ = sin⁻¹(2 × 0.2348) = sin⁻¹(0.4696) ≈ 28.0°.
• θ₃ = sin⁻¹(3 × 0.2348) = sin⁻¹(0.7044) ≈ 43.9°; separation = 43.9° - 28.0° = 15.9°.

Why C is correct:

• The grating equation d sin θ = m λ gives positions θ_m; the angle between m=2 and m=3 is θ₃ - θ₂ = 15.9°.

Why the others are wrong:

• A: Small-angle approximation θ ≈ (λ/d) × (180/π) ≈ 13.5°, ignores large-angle effects.
• B: Incorrect intermediate rounding of arcsin values.
• D: Approximates position of second-order maximum using small-angle for θ₂ ≈ 2(λ/d) × (180/π).

Final answer: C